A349687 Numbers whose numerator and denominator of their abundancy index are both Fibonacci numbers.

1, 2, 6, 15, 24, 26, 28, 84, 90, 96, 120, 270, 330, 496, 672, 1335, 1488, 1540, 1638, 8128, 24384, 27280, 44109, 68200, 131040, 447040, 523776, 18506880, 22256640, 33550336, 36197280, 38257095, 65688320, 91963648, 95472000, 100651008, 102136320, 176432256, 197308800

Offset: 1

Author

Amiram Eldar, Nov 25 2021

Keywords

• nonn

Comments

This sequence includes all the perfect numbers (A000396), 3-perfect numbers (A005820) and 5-perfect numbers (A046060).

The deficient terms, 1, 2, 15, 26, 1335, 44109, 38257095, ..., have an abundancy index which is a ratio of two consecutive Fibonacci numbers, 1/1, 3/2, 8/5, 21/13, 144/89, 610/377, 46368/28657, ..., which approaches the golden ratio phi = 1.618... (A001622) as the numerators and denominators get larger.

Examples

			2 is a term since sigma(2)/2 = 3/2 = Fibonacci(4)/Fibonacci(3).
15 is a term since sigma(15)/15 = 8/5 = Fibonacci(6)/Fibonacci(5).
		

Links

• Michel Marcus, 85 terms (some terms might be missing in this list).

Crossrefs

Cf. A000045, A010056, A000203, A001622, A017665, A017666.

Subsequences: A000396, A005820, A046060.

Similar sequences: A069070, A216780, A247086, A348658.

Programs

Mathematica

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[{5 n^2 - 4, 5 n^2 + 4}]; ai[n_] := DivisorSigma[1, n]/n; q[n_] := fibQ[Numerator[(ain = ai[n])]] && fibQ[Denominator[ain]]; Select[Range[10^6], q]

PARI

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
isok(n) = my(q=sigma(n)/n); isfib(numerator(q)) && isfib(denominator(q)); \\ Michel Marcus, Nov 25 2021